
"""simple docstring"""


from itertools import product






def    _snake_case					(		snake_case__ :  int						,   snake_case__ :  int  ):
	A     = sides_number
	A     = max_face_number * dice_number
	A     = [0] * (max_total + 1)

	A     = 1
	A     = range(snake_case__						,   max_face_number + 1  )
	for dice_numbers in product(snake_case__						,   repeat=snake_case__  ):
		A     = sum(snake_case__  )
		totals_frequencies[total] += 1

	return totals_frequencies






def    _snake_case					(		):
	A     = total_frequency_distribution(
	    sides_number=4						,   dice_number=9  )
	A     = total_frequency_distribution(
	    sides_number=6						,   dice_number=6  )

	A     = 0
	A     = 9
	A     = 4 * 9
	A     = 6
	for peter_total in range(snake_case__						,   max_peter_total + 1  ):
		peter_wins_count += peter_totals_frequencies[peter_total] * sum(
		    colin_totals_frequencies[min_colin_total:peter_total]  )

	A     = (4**9) * (6**6)
	A     = peter_wins_count / total_games_number

	A     = round(snake_case__						,   ndigits=7  )

	return rounded_peter_win_probability


if __name__ == "__main__":
		print(F"""{solution() = }""")
from __future__ import annotations


def 				lowerCAmelCase__       (			a__    ,     a__ = None    ,     a__ = None    ,     a__ = False    ,     )      ->tuple[int, float, str]:




  '''simple docstring'''

  _UpperCamelCase           =    cipher_alphabet or [chr(a__ ) for i in range(97    ,     123 )]

  # If the argument is None or the user provided an empty dictionary
  if not frequencies_dict:
    # Frequencies of letters in the english language (how much they show up)
    _UpperCamelCase           =    {
        "a": 0.08497,
        "b": 0.01492,
        "c": 0.02202,
        "d": 0.04253,
        "e": 0.11162,
        "f": 0.02228,
        "g": 0.02015,
        "h": 0.06094,
        "i": 0.07546,
        "j": 0.00153,
        "k": 0.01292,
        "l": 0.04025,
        "m": 0.02406,
        "n": 0.06749,
        "o": 0.07507,
        "p": 0.01929,
        "q": 0.00095,
        "r": 0.07587,
        "s": 0.06327,
        "t": 0.09356,
        "u": 0.02758,
        "v": 0.00978,
        "w": 0.02560,
        "x": 0.00150,
        "y": 0.01994,
        "z": 0.00077,
    }
  else:
    # Custom frequencies dictionary
    _UpperCamelCase           =    frequencies_dict

  if not case_sensitive:
    _UpperCamelCase           =    ciphertext.lower()

  # Chi squared statistic values
  _UpperCamelCase           =    {}

  # cycle through all of the shifts
  for shift in range(len(a__ ) ):
    _UpperCamelCase           =    ""

    # decrypt the message with the shift
    for letter in ciphertext:
      try:
        # Try to index the letter in the alphabet
        _UpperCamelCase           =    (alphabet_letters.index(letter.lower() ) - shift) % len(
            a__ )
        decrypted_with_shift += (
            alphabet_letters[new_key].upper()
            if case_sensitive and letter.isupper()
            else alphabet_letters[new_key]
        )
      except ValueError:
        # Append the character if it isn't in the alphabet
        decrypted_with_shift += letter

    _UpperCamelCase           =    0.0

    # Loop through each letter in the decoded message with the shift
    for letter in decrypted_with_shift:
      if case_sensitive:
        _UpperCamelCase           =    letter.lower()
        if letter in frequencies:
          # Get the amount of times the letter occurs in the message
          _UpperCamelCase           =    decrypted_with_shift.lower().count(a__ )

          # Get the excepcted amount of times the letter should appear based
          # on letter frequencies
          _UpperCamelCase           =    frequencies[letter] * occurrences

          # Complete the chi squared statistic formula
          _UpperCamelCase           =    ((occurrences - expected) ** 2) / expected

          # Add the margin of error to the total chi squared statistic
          chi_squared_statistic += chi_letter_value
      else:
        if letter.lower() in frequencies:
          # Get the amount of times the letter occurs in the message
          _UpperCamelCase           =    decrypted_with_shift.count(a__ )

          # Get the excepcted amount of times the letter should appear based
          # on letter frequencies
          _UpperCamelCase           =    frequencies[letter] * occurrences

          # Complete the chi squared statistic formula
          _UpperCamelCase           =    ((occurrences - expected) ** 2) / expected

          # Add the margin of error to the total chi squared statistic
          chi_squared_statistic += chi_letter_value

        # Add the data to the chi_squared_statistic_values dictionary
    _UpperCamelCase           =    (
        chi_squared_statistic,
        decrypted_with_shift,
    )

  # Get the most likely cipher by finding the cipher with the smallest chi squared
  # statistic
  def chi_squared_statistic_values_sorting_key(a__ ) -> tuple[float, str]:
    return chi_squared_statistic_values[key]

  _UpperCamelCase           =    min(
      a__    ,     key=a__    ,     )

  # Get all the data from the most likely cipher (key, decoded message)
  (
      (
      _UpperCamelCase
  )      ,						(
      _UpperCamelCase
  )      ,						
  )           =    chi_squared_statistic_values[most_likely_cipher]

  # Return the data on the most likely shift
  return (
      most_likely_cipher,
      most_likely_cipher_chi_squared_value,
      decoded_most_likely_cipher,
  )




